Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 136 / 2 / b / ii / Solution

The polynomialhas no root in ${\mathbb{F}}_3$ and is therefore irreducible. Definefor any fixed $c>1$. Its valuation ring has residue fieldThe completion at an irreducible polynomial over a finite field identifies the completion with ${\mathbb{F}}_{27}((T))$, where $T$ corresponds to the uniformizer $h(t)$.